How does the author determine the elements of order p or 4 in the group?

[moont14263]

My question is about the shaded area in the attachment?
How did the author get that all the elements of order p or 4 of L are contained in K? I mentioned the abstract but I do not think there is a need for that.

Help?

 

[micromass]

Take x\in L. Then there is a g in G such that x\in T^g and thus there is a y in T such that x=gyg^{-1}. If x has order p or order 4, then so does y. But by the previous sentence, y is contained in K. Since K is normal, we have that also x is in K.

 

[moont14263]

The symbol L=\bigcup_{g \in G} T^{g}, does it mean the union of sets or L=<T^{g},g \in G>and, if it the union of sets, then how did he gets that L equals to that union?